Differential Geometry Gymnastics [Japanese page]
Hold your arms as in Picture 1. Note that your palms are facing the
Lift up your arms as in Picture 2. Then your palms are facing the front.
Lower your arms to your both sides as in Picture 3. Your palms are
still facing the front.
Move your arms to your front as in Picture 4. Then your palms are
facing each other.
Though you did not twisted your arms in each step, your palms has been
turned 90 degrees.
If you repeat the steps as in Pictures 5, 6 and 7, then your palms will
be turned 90 degrees again and finally facing the heavens.
What is the reason of this phenomenon ?
Note that the tracks of your fingertips are spherical triangles each
with three right angles.
The reason exists in the fact that the sum of outer angles of such a
spherical triangle is 270 degrees.
It is well-known that the sum of outer angles of a polygon on a plane
is 360 degrees.
The difference 360 - 270 = 90 degrees appears as the twisting of your
If you try to draw other spherical polygon by your arm, then you will
find that the turning degree of your arm is proportional to the area of
that spherical polygon.
This is a consequence of so-called Gauss-Bonnet theorem in differential
In conclusion Differential Geometry Gymnastics is a way to feel that
the Gaussian curvature of a sphere is positive.
Thus one will be lead to the world of geometry and mathematics.
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